where, \(M_K\)  is the set of places of \(K\) , and for each \(v \in M_K\) , \(|.|_v\)  denotes the corresponding absolute values on \(K\)  satisfying the product formula, and can be extended to any algebraic closure and respective completions of \(K\) , so that \(h\)  can be well defined on \(\overline{K}\) . Also, we write \(K_v\)  for the completion of \(K\)  with respect to \(|.|_v\) , and we let \(\hat{K}_v\)  denote the completion of an algebraic closure of \(K_v\) .
Initially, we also recall that one can define the convergent limit \(\hat{h}_f(\alpha )=\lim _{n \rightarrow \infty } \dfrac{h(f(\alpha ))}{d^n}\)  for any \(\alpha \in K\) , called the canonical height associated with \(f\) , which also satisfies \(\hat{h}_f(f(\alpha ))=d\hat{h}_f(\alpha )\) , and that \(\alpha \)  has infinite orbit (is not preperiodic) if and only if \(\hat{h}_f(f(\alpha ))>0\) ( see [1]}).
